Hardy-Petrovitch-Hutchinson问题与偏θ函数
Hardy-Petrovitch-Hutchinson's problem and partial theta function
摘要：
在1907年M.彼得罗维奇发起了研究一类整个职能所有的有限部分是真正的根源多项式。一个明确的描述这一类的一系列的系数是不可能的，因为它是由一个无限数量的判别不等式，每一个度。然而，有趣的必要或充分条件可以公式化。特别地，J. I. Hutchinson已经证明了整函数p（x）=a_0+a_1x+.+ a_nx^n+.具有严格正系数的有限线段a_ix^i+.+ a_jx^j有所有实根当且仅当对所有i= 1，2，.其中a_1 ^2/ a_{i-1} a_{i+1}大于或等于4。在本文中，我们给出了M. Petrovitch所考虑的类的比率a_i^2/ a_{i-1} a_{i+1}的精确下界。特别是，我们表明，这些最小值的限制，当i趋于无穷大等于反的最大正值的参数，其中经典的部分θ函数属于Laguerre-Polya类。
英文摘要：
In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminantal inequalities one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J.I.Hutchinson has shown that an entire function p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the property that any its finite segment a_ix^i+...+a_jx^j has all real roots if and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or equal to 4. In the present paper we give sharp lower bounds on the ratios a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular, we show that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.
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